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Metamath Proof Explorer

Theorem bj-19.37im

Description: One direction of 19.37 from the same axioms as 19.37imv . (Contributed by BJ, 2-Dec-2023)

		Ref	Expression
	Assertion	bj-19.37im	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )

Step	Hyp	Ref	Expression
1		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		bj-nnfa	⊢ ( Ⅎ' 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3	2	imim1d	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )
4	1 3	biimtrid	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )